Enumeration of trees from finite number of nodes

ABSTRACT

Embodiments of methods, apparatuses, devices and/or systems for manipulating hierarchical sets of data are disclosed.

RELATED APPLICATION

This disclosure claims priority pursuant to 35 USC 119(e) from U.S. Provisional Patent Application Ser. No. 60/632,203, filed on Nov. 30, 2004, by Schiffmann, et al., titled, “ENUMERATION OF TREES FROM FINITE NUMBER OF NODES,” assigned to the assignee of the presently claimed subject matter.

BACKGROUND

This disclosure is related to hierarchical data arrangements and, more particularly, to manipulating such data arrangements.

In a variety of fields, data or a set of data, may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships between particular pieces of data and the like. However, manipulating such data representations is not straight-forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, one example may include a relational database. Techniques for performing operations on such a database, for example, are computationally complex or otherwise cumbersome. A continuing need, therefore, exists for additional techniques for manipulating data hierarchies.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in the concluding portion of the specification. The claimed subject matter, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference of the following detailed description when read with the accompanying drawings in which:

FIG. 1 is a schematic diagram of one embodiment of a tree;

FIG. 2 is a schematic diagram illustrating one embodiment of a binary edge labeled tree (BELT);

FIG. 3 is a schematic diagram illustrating another embodiment of a BELT;

FIG. 4 is a table illustrating a particular embodiment of an association between natural numerals and BELTs;

FIG. 5 is a table illustrating a portion of the Kleene enumeration of non-composite numerals;

FIG. 6 is a table illustrating one embodiment of a linear notation for representing a graphical depiction of a hierarchical set of data;

FIG. 7 is a schematic diagram of an embodiment of a node labeled tree;

FIG. 8 is a schematic diagram illustrating another embodiment of a node labeled tree;

FIG. 9 is a schematic diagram illustrating a generalized representation of an embodiment of one or more subtrees according to an embodiment.

FIGS. 10 a, 10 b, 11 a and 11 b are schematic diagrams illustrating applications of one embodiment of a push operation according to an embodiment.

FIGS. 12 through 16 are schematic diagrams illustrating allocations of five nodes among subtree slots for trees having five nodes according to an embodiment.

FIG. 17 is a flow diagram illustrating a process to enumerate all trees which are configurable from a finite number of nodes according to an embodiment.

FIG. 18 is a schematic diagram illustrating a distribution of one unique allocation of a single node depending from a root node.

FIG. 19 is a schematic diagram illustrating BELTs that may be configured from two nodes.

FIGS. 20 through 22 are schematic diagrams illustrating BELTs that may be configured from three nodes.

FIGS. 23 through 25 are schematic diagrams illustrating BELTs that may be configured from four nodes.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the claimed subject matter. However, it will be understood by those skilled in the art that the claimed subject matter may be practiced without these specific details. In other instances, well-known methods, procedures, components and/or circuits have not been described in detail so as not to obscure the claimed subject matter.

Some portions of the detailed description which follow are presented in terms of algorithms and/or symbolic representations of operations on data bits or binary digital signals stored within a computing system memory, such as a computer memory. These algorithmic descriptions and/or representations are the techniques used by those of ordinary skill in the data processing arts to convey the substance of their work to others skilled in the art. An algorithm is here, and generally, considered to be a self-consistent sequence of operations and/or similar processing leading to a desired result. The operations and/or processing involve physical manipulations of physical quantities. Typically, although not necessarily, these quantities may take the form of electrical and/or magnetic signals capable of being stored, transferred, combined, compared and/or otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, data, values, elements, symbols, characters, terms, numbers, numerals and/or the like. It should be understood, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels. Unless specifically stated otherwise, as apparent from the following discussion, it is appreciated that throughout this specification discussions utilizing terms such as “processing”, “computing”, “calculating”, “determining” and/or the like refer to the actions and/or processes of a computing platform, such as a computer or a similar electronic computing device, that manipulates and/or transforms data represented as physical electronic and/or magnetic quantities within the computing platform's memories, registers, and/or other information storage, transmission, and/or display devices.

In a variety of fields, data or sets of data may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships between particular pieces of data and the like. However, manipulating such data representations is not straight forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, one example may include a relational data base. Techniques for performing operations on such a data base for example, may be computationally complex or otherwise cumbersome. A continuing need, therefore, exists for additional techniques for manipulating data hierarchies.

As previously discussed, in a variety of fields, it is convenient or desirable to represent data, a set of data and/or other information in a hierarchical fashion. In this context, such a hierarchy of data shall be referred to as a “tree.” In a particular embodiment, a tree may comprise a finite, rooted, connected, unordered, acyclic graph. This is illustrated here, for example, in FIG. 1 by embodiment 100. As illustrated, the root of this particular embodiment encompasses node 105. In addition to 105, there are eight other nodes designated 110 to 140, respectively. Likewise, the nodes are connected by branches referred to, in this context, as edges. Thus, the nodes of this tree are connected by eight edges. This embodiment, therefore, illustrates a finite tree that is rooted by node 105. Furthermore, the nodes are connected, meaning, in this context, that a path exists between any two nodes of the tree. The tree is likewise acyclic, meaning here, that no path in the tree forms a complete loop. Here, unordered refers to the notion that there is no implied ordering or precedence among nodes attached to a common node, despite the appearance of ordering in a graphical illustration.

As previously suggested, in a variety of contexts, it may be convenient and/or desirable to represent a hierarchy of data and/or other information using a structure, such as the embodiment illustrated in FIG. 1. One particular embodiment, without loss of generality, of a tree may include edges that are labeled with data and/or other values. Likewise, in one particular embodiment, such data or values may be limited to binary data, that is, in this example, either a binary one or a binary zero. Here, such an embodiment may be referred to as a binary edge labeled tree (BELT), as shall be discussed in more detail hereinafter.

One example of a BELT is illustrated by embodiment 200 of FIG. 2. Thus, as illustrated, the edges of the BELT shown in FIG. 2 are labeled with either a binary zero or binary one. FIG. 3 illustrates another embodiment 300 of a different binary edge labeled tree. It is noted that this tree is similar to the embodiment of FIG. 2. Without belaboring the present discussion, additional descriptions of how BELTs may represent a hierarchy of data may be found in U.S. Provisional Patent Application No. 60/543,371, filed on Feb. 9, 2004, by J. J. LeTourneau, titled, “Manipulating Sets of Hierarchical Data,” assigned to the assignee of the presently claimed subject matter with reference to FIGS. 4-7 therein.

Binary edge labeled trees may also be enumerated. Thus, for this particular embodiment, although the claimed subject matter is not limited in scope in this respect, a method of enumerating a set of trees begins with enumeration of an empty binary edge labeled tree and a one node binary edge labeled tree. Here, the empty tree is associated with the zero and has a symbolic representation as illustrated in FIG. 4 (circle). Likewise, the one node tree, which holds no data, is associated with the one and has a graphical representation of a single node. For higher positive natural numerals, however, this embodiment of a method of enumerating a set of trees comprises positioning a tree at location k, k being a positive greater than three, where k is the product of u and v, u and v comprising positive s greater than one, such that the tree is formed by a union of the trees at positions u and v. Likewise, for those locations that are not a product of other natural positive numerals greater than one, that is, for locations that comprise non-composite numerals, denoted here by j, for example, j being a positive natural numeral greater than one, a tree is positioned at location j such that the tree is formed by finding the first tree in the prior enumeration such that the binary edge labeled tree obtainable from this first tree by attaching a node to the particular tree as a new root node and labeling the edge between the new root node and the prior root node with a binary “0” label is not in the enumeration at some position lower than j; however, if the binary edge labeled tree obtainable from that first tree, as just described, is present in the enumeration with a binary “0” label for the new edge, but not with a binary “1” label, then the tree at position j is that tree with a binary “1” label for the new edge. This may be illustrated, for example in FIG. 4, as described in more detail below.

However, for this particular embodiment, although the claimed subject matter is not limited in scope in this respect, a method of enumerating a set of ordered trees may begin with enumeration of an empty binary edge labeled tree and a one node binary edge labeled tree. Thus, the empty tree is associated with the zero and has a symbolic representation as illustrated in FIG. 4 (circle). Likewise, the one node tree, which holds no data, is associated with the one and has a graphical representation of a single node. For higher positive natural numerals, ordered trees may be generated by a process described, for example, in “The Lexicographic Generation of Ordered Trees,” by S. Zaks, The Journal of Theoretical Computer Science, Vol. 10(1), pp. 63-82, 1980, or Enumerating Ordered Trees Lexicographically,” by M. G. Er, Computation Journal, Vol. 28, Issue 5, pp. 538-542, 1985.

As illustrated, for this particular embodiment, and as previously described, the empty tree has zero nodes and is associated with the zero. Likewise, the one node tree root comprises a single node and is associated with the one. Thus, to obtain the tree at position two, a root node is attached and connected to the prior root node by an edge. Likewise, here, by convention, the edge is labeled with a binary zero. If, however, the tree formed by the immediately proceeding approach were present in the prior enumeration of trees, then a similar process embodiment is followed, but, instead, the new edge is labeled with a binary one rather than a binary zero. Thus, for example, in order to obtain the binary edge labeled tree for position three, a new root node is connected to the root node by an edge and that edge is labeled with a binary one.

Continuing with this example, to obtain the binary edge labeled tree for position four, observe that numeral four is the product of numeral two times numeral two. Thus, a union is formed at the root of two trees, where, here, each of those trees is associated with the positive natural numeral two. Likewise, to obtain the binary edge labeled tree for position five, begin with the binary edge labeled tree for position two and follow the previously articulated approach of adding a root and an edge and labeling it with a binary zero.

In this context, adding a root node and an edge and labeling it binary zero is referred to as a “zero-push” operation and adding a root node and an edge and labeling it binary one is referred to as a “one-push” operation. Based at least in part on the prior description, for this particular embodiment, it may now be demonstrated that if k is any positive natural numeral and a tree x is positioned at location k, then a non-composite numeral is associated with the zero-push of that tree and a non-composite numeral is associated with the one-push for that tree. Furthermore, the non-composite index of the zero-push of the tree comprises 2k−1, whereas the non-composite index of the one-push of the tree comprises 2k, where the index corresponds to the argument of the well-known Kleene enumeration on positive natural numerals of non-composite numerals, as illustrated, for example, in part in FIG. 5. Thus, referring again to FIG. 4, the one-push of the root tree is the tree at position three. This follows from FIG. 5 since P(2*1)=P(2)=3. Likewise, the tree at position five is the zero-push of the tree at position 2. Again, this follows from FIG. 5 since P(2*2−1)=P(3)=5.

In this context, the approach just described may be referred to as vectorizing non-composite numerals. In the embodiment just described, this was accomplished in pairs, although, of course, the claimed subject matter is not limited in scope in this respect. This may be accomplished in any number of numeral combinations, such as triplets, quadruplets, etc. Thus, using a quadruplet example, it is possible to construct trees such that if k is any positive natural numeral and a tree x is positioned at location k, then a non-composite numeral is associated with the zero-push of that tree, a non-composite numeral is associated with the one-push for that tree, a non-composite numeral is associated with the two-push for that tree, and a non-composite number is associated with the three-push for that tree. Furthermore, the index of the non-composite numeral is such that for a zero-push of the tree, the index comprises (4k−3), for a one-push of a tree, the index comprises (4k−2), for a two-push of a tree, the index comprises (4k−1), and for a three-push of a tree the index comprise (4k), where the index corresponds to the Kleene enumeration of non-composite numerals, P(index), such as provided in FIG. 5.

In the previously described enumeration of binary edged labeled trees, a mechanism may be employed to reduce or convert complex manipulations of hierarchical data to multiplication of natural numerals. For example, if it is desired to combine, or merge at their roots, two trees of hierarchical data, a complex task both computationally and graphically, instead, for this particular embodiment, the two trees may be converted to numerical data by using the previously described association embodiment between binary edge labeled trees and natural numerals. The resulting numerical data from the prior conversion may then be multiplied, and the resulting product may then be converted to a binary edge labeled tree by using a table look up of the previously described association embodiment. It is noted that a subtle distinction may be made between an enumeration embodiment and an association embodiment. Enumeration may comprise listing, in this example, a particular ordered embodiment of BELTs, whereas an association provides a relationship between, in this example, a particular ordered embodiment of BELTs and natural numerals. It is, of course, appreciated that many different enumeration and association embodiments may be employed to execute the operations discussed above and hereinafter, and the claimed subject matter is intended to cover all such enumeration and association embodiments.

Likewise, a process embodiment that is a reversal to the previously described embodiments may also be employed. Thus, complex hierarchies of data may be split or divided, when this is desired. For example, a binary edge labeled tree to be divided may be converted to a piece of numerical data, such as by using the previously described association embodiment. This data may then be factored into two pieces of numerical data whose product produces the previously mentioned piece of numerical data. These two pieces of numerical data may then be converted to trees, again, by using the prior association embodiment, for example.

Another form of manipulating hierarchical sets of data may involve ordering or hashing. This may be desirable for any one of a number of different operations to be performed on the sets of data. One approach is similar to the previously described embodiment. For example, it may be desired to order a given set of trees. Doing so may involve converting the trees to numerical data, as previously described, using an association embodiment. The numerical data may then be ordered and the numerical data may then be converted back to binary edge labeled trees using the previously described association embodiment, or an alternate association embodiment, for example.

It is noted that there may be any one of a number of different ways of converting from numerals or numerical data values to a binary edge labeled tree or from a binary string to a binary edge labeled tree, and vice-versa. Nonetheless, a convenient method for doing so with this particular embodiment includes storing a table providing an association embodiment between natural numerals, binary strings and binary edge labeled trees, such as the embodiment previously described. Thus, once it is desired to convert from one to the other, such as from a binary string to a BELT, from a natural numeral to a BELT, or vice-versa, for example, a table look up operation may be performed using the association embodiment.

Techniques for performing table look ups are well-known and well-understood. Thus, this will not be discussed in detail here. However, it shall be appreciated that any and all of the previously described and/or later described processing, operations, conversions, transformations, manipulations, etc. of strings, trees, numerals, data, etc. may be performed on one or more computing platforms or similar computing devices, such as those that may include a memory to store a table as just described, although, the claimed subject matter is not necessarily limited in scope to this particular approach. Thus, for example, a hierarchy of data may be formed by combining two or more hierarchies of data, such as by applying a previously described embodiment. Likewise, multiple hierarchies of data may be formed by splitting or dividing a particular hierarchy of data, again, such as by applying a previously described embodiment. Likewise, additional operations and/or manipulations of data hierarchies may be performed, such as ordering hierarchies of data and more. It is intended that the claimed subject matter cover such embodiments.

Much of the prior discussion was provided in the context of binary edge labeled trees. Nonetheless, as alluded to previously, binary edge labeled trees and binary node labeled trees may be employed nearly interchangeably to represent substantially the same hierarchy of data. In particular, a binary node labeled tree may be associated with a binary edge labeled tree where the nodes of the binary node labeled tree take the same values as the edges of the binary edge labeled tree, except that the root node of the binary node labeled tree may comprise a node having a zero value or a null value. Thus, rather than employing binary edge labeled trees, the previously described embodiments may alternatively be performed using binary node labeled trees. As one example embodiment, operations and/or manipulations may be employed using binary edge labeled trees and then the resulting binary edge labeled tree may be converted to a binary node labeled tree. However, in another embodiment, operations and/or manipulations may be performed directly using binary node labeled trees where a different association embodiment, that is, in this example, one that employs binary node labeled trees, is employed.

In accordance with the claimed subject matter, therefore, any tree, regardless of whether it is binary edge labeled, binary node labeled, non-binary, a feature tree, or otherwise, may be manipulated and/or operated upon in a manner similar to the approach of the previously described embodiments. Typically, different association embodiments shall be employed, depending at least in part, for example, upon the particular type of tree. For example, and as shall be described in more detail below in connection with FIG. 7, a node labeled tree in which the nodes are labeled with natural numerals or data values may be converted to a binary edge labeled tree. Furthermore, this may be accomplished with approximately the same amount of storage. For example, for this particular embodiment, this may involve substantially the same amount of node and/or edge data label values.

As previously noted, the claimed subject matter is not limited in scope to this particular example, however, as illustrated in more detail hereinafter, the tree illustrated in FIG. 7 is converted to a binary edge labeled tree through a sequence of processing depicted here as graph operations, although such a conversion may alternatively be implemented by operations implemented otherwise, one such example being a computing platform, for example. Alternatively, it may be desirable, depending upon the particular embodiment, to convert trees to, for example binary node labeled trees. Likewise, other embodiments in which trees of one form are converted to trees of another form are also included within the scope of the claimed subject. However, for this particular embodiment, it will be assumed that the association between trees and numerals, such as previously described, is depicted or enumerated in terms of binary edge labeled trees, as previously illustrated, for example. Thus, in this example, a particular tree, embodiment 1100, is illustrated in FIG. 7, comprises a node labeled tree rather than an edge labeled tree. Without belaboring the present discussion, a process of converting a node labeled tree such as that illustrated in FIG. 7 to a BELT may be found in U.S. Provisional Patent Application No. 60/543,371, filed on Feb. 9, 2004, by J. J. LeTourneau, titled, “Manipulating Sets of Hierarchical Data,” assigned to the assignee of the presently claimed subject matter with reference to FIGS. 11-16 therein.

In another embodiment, however, a particular tree may include null types or, more particularly, some node values denoted by the empty set. This is illustrated, for example, by the tree in FIG. 13, although, of course, this is simply one example. An advantage of employing null types includes the ability to address a broader array of hierarchical data sets. For example, without loss of generality and not intending to limit the scope of the claimed subject matter in any way, a null type permits representing in a relational database, as one example, situations where a particular attribute does not exist. As may be appreciated, this is different from a situation, for example, where a particular attribute may take on a numeral value of zero. Thus, it may be desirable to be able to address both situations when representing, operating upon and/or manipulating hierarchical sets of data. A tree with nulls may be converted to a tree without nulls as described in U.S. Provisional Patent Application No. 60/543,371, filed on Feb. 9, 2004, by J. J. LeTourneau, titled, “Manipulating Sets of Hierarchical Data,” assigned to the assignee of the presently claimed subject matter with reference to FIGS. 17-22 therein.

Likewise, in an alternative embodiment, a node labeled tree may comprise fixed length tuples of numerals. For such an embodiment, such multiple numerals may be combined into a single numeral, such as by employing Cantor pairing operations, for example. See, for example, Logical Number Theory, An Introduction, by Craig Smorynski, pp, 14-23, available from Springer-Verlag, 1991. This approach should produce a tree to which the previously described embodiments may then be applied. Furthermore, for one embodiment, a tree in which nodes are labeled with numerals or numerical data, rather than binary data, may be converted to a binary edge labeled tree and/or binary node labeled tree, and, for another embodiment, a tree in which edges are labeled with numerals or numerical data, rather than binary data, may be converted to a binary edge labeled tree and/or binary node labeled tree.

Furthermore, a tree in which both the nodes and the edges are labeled may be referred to in this context as a feature tree and may be converted to a binary edge labeled tree and/or binary node labeled tree. For example, without intending to limit the scope of the claimed subject matter, in one approach, a feature tree may be converted by converting any labeled node with its labeled outgoing edge to an ordered pair of labels for the particular node. Using the embodiment described above, this tree may then be converted to a binary edge labeled tree.

In yet another embodiment, for trees in which data labels do not comprise simply natural numerals, such as, as one example, trees that include negative numerals, such data labels may be converted to an ordered pair of numerals. For example, the first numeral may represent a data type. Examples include a data type such as negative, dollars, etc. As described above, such trees may also be converted to binary edge labeled trees, such as by applying the previously described embodiment, for example.

As previously described, trees may be employed to graphically represent a hierarchy of data or a hierarchy of a set of data. This has been illustrated in some detail for binary edge labeled trees, for example. As the previous figures, illustrate, however, such graphical hierarchical representations typically employ two spatial dimensions to depict the relationship among different pieces of data. This may be disadvantageous in some situations where a one dimensional representation or arrangement of symbols, such as is employed with alphabetic letters, for example, that are combined to create a linear collection of successive symbols or notations, such as words, would be more convenient.

FIG. 6 is a table that illustrates one particular embodiment of employing symbols, concatenated along one spatial dimension, here from left to right, by convention, to represent such a data hierarchy. The table includes four columns. The first column denotes natural numerals. The second column denotes binary strings. The third column denotes a one dimensional arrangement of symbols employed to represent the binary edge labeled trees for that particular position. Of course, as previously described in connection with prior embodiments, the claimed subject matter is not limited in scope to binary strings or binary edge labeled trees. Thus, in alternative embodiments, any tree may be represented with symbols organized in a one dimensional arrangement, as is demonstrated with this particular embodiment.

According to an embodiment, a finite number of trees may be enumerated from a finite number of nodes. For each tree enumerated from the finite number of nodes, a natural numeral may be associated with the enumerated tree based, at least in part, on an association between trees and natural numerals. For example, a set of fully configured, unordered trees may be enumerated from a finite number of nodes. The enumerated trees may then be represented by a corresponding set of natural numerals.

For this particular embodiment, a tree may be expressed as one or more “subtrees” coupled at the root node of the tree. A subtree is coupled to the root node of the tree by an edge and independently has properties of a tree, except that the subtree is part of a larger tree. For example, here, a subtree comprises at least a “root” node coupled by an edge to a root node of the tree. Additional nodes and edges may be coupled to the root node of the subtree. While a subtree may be coupled to a root node of a tree by an edge, the size and shape of the subtree may express information like that of a tree having the same size and shape as the subtree. The subtrees coupled together at the root node of a tree may be referred to as “subtree children” of the root node where a subtree may be referred to as a “subtree child” of the tree in this embodiment.

FIG. 9 is a schematic diagram illustrating a representation an embodiment of one or more subtrees connected via an edge having a label “e” with a subtree root node having a label “n.” A value associated with the one or more subtrees may be represented by “x” which may represent information expressed as a natural numeral as illustrated above with reference to FIG. 4. However, this is merely an example of a representation of a subtree and the claimed subject matter is not limited in this respect. Accordingly, the value of the tree shown in FIG. 9 may be expressed as a push of the subtree having the value x. A push operation on the subtree may be represented in relation (1) as follows:

<j,k>—push(x)=P[kx+j−k+(2−r)], if j<k and k>0  (1)

where:

-   -   P(m)=Kleene enumeration function for generating a sequence of         non-composite numbers illustrated with reference to FIG. 5;     -   k=total number of values possible for a label;     -   j=actual computed label index value;     -   x=value of “pushed” subtree with edge “e” and root label “n”;     -   r=defined value of tree system root/singleton node (either 0 or         1).

It should be noted that “j” is a function of the specific values of “e” (the specific edge label) and “n” (the specific node label). It should be understood that while the push operation of relation (1) is suitable for performing specific embodiments described herein, this push operation is merely an example of how a push operation may be performed on a subtree and the claimed subject matter is not limited in this respect.

In addition to applying the push operation to a single tree or subtree having a value x, the push operation may be applied to multiple tree or subtree elements of a set in relation (2) as follows:

<j,k>—push[{a,b,c}]={<j,k>—push(a)}U{<j,k>—push(b)}U{<j,k>—push(c)}  (2)

where a, b and c are numerical representations of tree or subtree elements in the pushed set. The result of the operation of relation (2) may be referred to as a “pushed set” of tree or subtree elements.

FIGS. 10 a, 10 b, 11 a and 11 b illustrate applications of the push operation of relation (1) to specific subtrees. FIG. 10 a shows a BELT having a value of “2”. As such, x=2, k=2 and r=1. FIG. 10 b illustrates the result of a push of the tree in FIG. 10 a by an edge having a label zero (i.e., a zero-push). The value of j for this push operation is zero. Accordingly, the push operation provides a numeral associated with the pushed BELT as follows:

<j,k>—push(x)=P[2*2+0−2+2−1]=P[3]=5.

FIG. 11 a shows a structure-only tree (i.e., a tree with unlabeled elements) with two nodes having a value of x=2. As such, k=1 (since there are no labels) and r=1. FIG. 11 b illustrates the result of a push of the tree in FIG. 11 a by an unlabeled edge. The value of j is zero since there are no labels. Accordingly, the push operation determines a numeral associated with the pushed unlabeled tree as follows:

<j,k>—push(x)=P[1*2+0−1+2−1]=P[2]=3.

A number of subtrees coupled to a root node of a tree may be configured from one or more “subtree slots.” In this embodiment, a subtree slot represents a possible location of a subtree coupled to the root node of a larger tree. For a tree enumerated from a finite number N of nodes, a first node may serve as a root node while the remaining N−1 nodes may be configured into subtrees coupled to the root node. Up to N−1 subtrees may be formed from the remaining N−1 nodes (where each subtree contains a single node coupled to the root node by an edge). Accordingly, N−1 subtree slots may be identified for the set of trees enumerated from N nodes.

For any particular enumerated tree, according to an embodiment, an “arrangement of subtree slots” sets forth an unordered set of subtree slots having a subtree of at least one node. In enumerating a tree from N nodes, for example, one arrangement of subtree slots having a subtree in each of N−1 subtree slots yields a tree having a single node in each of N−1 subtree slots coupled to the root node by an edge. In another arrangement of subtree slots for a tree enumerated from N nodes, for example, a single subtree slot may contain all N−1 nodes. However, these are merely examples of arrangements of subtree slots and that other arrangements may be possible.

A subtree slot in an arrangement of subtree slots may have one or more nodes according to one or more “allocations of nodes” among the subtree slots defining a number of nodes in the subtree slot of the arrangement. In other words, such an allocation of nodes for an arrangement of subtree slots (i.e., the number of subtree slots having at least one node) sets forth the number nodes being allocated to each subtree slot. The nodes allocated to a subtree slot (e.g., according to an allocation of nodes among subtrees) may be set out in one or more possible, unordered configurations of a subtree where a possible configuration may represent a set of data and/or other information in a hierarchy of data using properties of a tree. In a particular embodiment, a configuration of a subtree may comprise a finite, rooted, connected, unordered acyclic graph as illustrated with reference to FIGS. 1-8.

According to an embodiment, a finite number of possible trees or subtrees may be enumerated from a finite number of N nodes. For any tree (or subtree) with N such nodes, there are N−1 available arrangements of subtree children. That is, for a tree (or subtree) with N nodes there may be any arrangement of zero to N−1 subtree children coupled to a root node. Accordingly, an arrangement of subtree children may define up to N−1 subtree slots that may be coupled or merged at a root node of the tree. For any particular arrangement of subtree children (e.g., represented as a count of the number of subtree slots with at least one node coupled to a root node) an allocation of nodes may be expressed as an unordered set containing N−1 elements where an element of the unordered set defines 0 to N−1 nodes for a corresponding subtree slot. An element of the unordered set may represent a count (ranging from 0 to N−1) of a number of nodes being allocated to the corresponding subtree slot. The sum of the counts represented by the elements may then equal N−1.

For any particular allocation of nodes among one or more subtree slots in an arrangement of subtree slots, there may be one or more possible “fully configured trees” defining a relationship among the allocated nodes using each of the allocated nodes. Such a fully configured tree may represent hierarchical data in a finite, rooted, connected, unordered, acyclic graph. Similarly, for an allocation of nodes to a subtree slot, the fully configured tree may comprise a corresponding subtree child representing hierarchical data in a finite, rooted, connected, unordered, acyclic graph. Notation “{FTs: N}” provides a shorthand notation for this particular embodiment to indicate the set of all fully configured trees that may possibly be configured from exactly N nodes under a given set of conditions (e.g., parameters characterizing the type of tree such as k, j, and r as described above in connection with the push operation of relation (1)) Of course, this is merely an example of how a set of fully configured trees from exactly N nodes may be characterized and the claimed subject matter is not limited to this particular approach.

However, continuing with this example, as illustrated in FIGS. 12 through 16, for example, a tree with five nodes may have one to four subtrees where each subtree includes at least one of the five nodes. FIG. 12 shows a single subtree with four nodes, FIGS. 13 and 15 two subtrees, FIG. 16 shows three subtrees and FIG. 14 shows four subtrees.

For any particular arrangement of subtrees with a finite number of nodes, the arrangement may be characterized as having one or more possible allocations of the nodes among the subtrees in the arrangement. Table 1 below illustrates five allocations of five nodes among subtrees among arrangements of one to four subtree slots.

TABLE 1 Allocation of Nodes (A_(i)) Among Subtree Slots Allocation Subtree Slots (m) Number (i) 0 1 2 3 0 4 0 0 0 1 3 1 0 0 2 2 2 0 0 3 2 1 1 0 4 1 1 1 1

For this embodiment, an allocation defines a set of unordered elements. Thus, it should be understood that the ordering of allocations of specific quantities of nodes to particular subtree slots (numbered 0 to 3) outlined in Table 1 is arbitrary and that there are other ways to sequence the allocation of nodes among subtree slots. For this particular example, allocation 0 indicates the allocation of four nodes to a single subtree slot 4. The allocation merely represents the allocation of four nodes to a single subtree slot, irrespective of the particularly numbered subtree slot. Accordingly, duplicate entries of four nodes being allocated to either of subtree slot 1, 2 or 3 are not represented in Table 1. Similarly, allocation 2 of this particular example indicates the allocation of two nodes to subtree slot 0 and two nodes to subtree slot 1. This allocation merely represents the allocation of two nodes to a first subtree slot and two nodes to a second, different, subtree slot. Accordingly, duplicate entries of two nodes being allocated to each of slots 2 and 3, among others, are not represented in Table 1.

According to one embodiment, a merger operation discussed above (for combining trees at their root nodes to provide a graphical and numerical expression of the resulting merged tree) may be expanded to create a set merger operation to include a merger among trees in different sets of trees. Here, a member tree of a first set merges with a member tree of a second set to provide a third, merged set containing the merged trees as elements, for all members of both sets. Regarding the representation of the trees as natural numerals, the set merger operation to merge sets X and Y may be expressed as follows:

$\begin{matrix} \begin{matrix} {{XxY} = {{\left\{ {x_{1},x_{2},x_{3},\ldots \mspace{14mu},x_{n}} \right\} x\left\{ {y_{1},y_{2},{y_{3}\ldots}\mspace{14mu},y_{m}} \right\}} =}} \\ {= \left\{ {{x_{1}*y_{1}},{x_{1}*y_{2}},{x_{1}*y_{3}},\ldots \mspace{14mu},{x_{1}*y_{m}}} \right\}} \\ {{U\left\{ {{x_{2}*y_{1}},{x_{2}*y_{2}},{x_{2}*y_{3}\ldots}\mspace{14mu},{x_{1}*y_{m}}} \right\}}} \\ {{U\left\{ {{x_{3}*y_{2}},{x_{3}*y_{2}},{x_{3}*y_{3}\ldots}\mspace{14mu},{x_{3}*y_{m}}} \right\}}} \\ {{U\left\{ {{x_{n}*y_{1}},{x_{n}*y_{2}},{x_{n}*y_{3}\ldots}\mspace{14mu},{x_{n}*y_{m}}} \right\}}} \end{matrix} & (3) \end{matrix}$

where:

x₁, x₂, x₃, . . . x_(n) are the natural numerals representing trees in set X; and

y₁, y₂, y₃, . . . y_(m) are the natural numerals representing trees in set Y;

Using the set merger operation illustrated at relation (3), for this particular embodiment, the merger of sets {FTs: N} and {FTs: M}, representing all trees which may be enumerated from N and M nodes, respectively, provides a set of trees enumerating all trees formed by the merger of any tree in {FTs: N} with any tree in {FTs: M}.

FIG. 17 is a flow diagram illustrating an embodiment of a process 3200 for enumerating trees which may possibly be configured from an N number of nodes {FTs: N} (under a given set of conditions) and determining a corresponding unique natural numerals for representing the particular enumerated trees. The process 3200 may be implemented as a computer program or subroutine expressed as machine-readable instructions which are executable by a processor. However, the claimed subject matter is not limited in scope in this respect.

Block 3202 defines parameters indicating the size and type of trees that are to be enumerated. N determines the number of nodes in the enumerated trees. The parameters r and k determine the defined value of the enumerated tree system and total number of values possible for a label as indicated for the push operation of relation (1). Block 3204 initializes full tree set {FTs: 1}={1} to represent a natural numeral for a single root node. As discussed below, in this particular example, the enumerated trees are generated from applying a push operation of relation (1) to the single root node. It should be understood, however, that the single root node may be associated with a different natural numeral and the claimed subject matter is not limited in this respect. Block 3204 also initializes full tree set {FTs: N}=Ø to indicate a set that initially contains no elements to which elements are to be added through the subsequent execution of blocks 3206 through 3220.

Block 3206 determines an allocation A_(i) of nodes among up to N−1 subtree slots as illustrated in the example of Table 1 above. In the example of Table 1, allocation A_(i) for i=0 to 4 represents an allocation of four nodes among up to four subtrees. For allocations A_(i), a loop defined by blocks 3208 through 3218 determines a set containing trees that may be enumerated for the allocation A_(i) (denoted as “{FTs: A_(i)}”) and an associated natural numeral for the particular enumerated tree in {FTs: A_(i)}. A subtree slot m (where m=0 to N−1) in A_(i) includes a number of nodes n_(m) (from 0 to N−1). For a subtree slot in A_(i), a loop defined by blocks 3210 through 3214 enumerates the set of all fully configured trees that may be configured from n_(m) nodes {FTs: n_(m)}. Block 3212 may determine {FTs: n_(m)} by recursively executing the described process 3200 using the argument of n_(m) as N in block 3202 of the recursively executed process 3200 (while maintaining the parameters k and r in block 3202 of the recursively executed process 3200).

Block 3216 determines the set {FTs: A_(i)} based, at least in part, upon the sets {FTs: n_(m)}, m=0 to N−1, determined in the loop of blocks 3210 through 3214. For a set {FTs: n_(m)}, block 3216 determines the push of this set according to the push operation of relation (2). As discussed above with reference to FIG. 9, a result of the push operation of relation (1) may depend on particular information associated with edge or node labels coupling the pushed subtree to a root node. In the case of a BELT, for example, the pushed subtree may be coupled to the root node by an edge labeled one or zero. Thus, in this particular example, to enumerate all trees resulting from a pushed binary edge labeled subtree, the push operation of relation (1) may be applied with j=0 (e.g., a zero-push) to enumerate a first resulting tree and with j=1 (e.g., a one-push) to enumerate a second resulting tree. Similarly, in the application of the push operation of relation (2) to a set of tree elements, the resulting enumerated pushed tree element may depend on particular information associated with edge or node labels coupling the pushed tree element to a root node.

Thus, in this particular example, to enumerate all trees resulting from the application of push operation of relation (2) to a set of tree elements, block 3216 provides a union of j-push{FTs: n_(m)} sets for all possible values of j (e.g., for all j=0 to k). In the case of enumerating all BELTs from N nodes, for example, the operation “Push{FTs: n_(m)}” as shown in block 3216 refers to the union of zero-push{FTs: n_(m)} and one-push{FTs: n_(m)}. For the enumeration of trees with edge labels having more than two possible values, however, the operation “Push{FTs: n_(m)}” may refer to the union of more than two such pushed sets (i.e., a union of pushed sets for each possible value for j). Block 3216 then merges the pushed sets {FTs: n_(m)} using the set merger operation as shown in relation (3). Block 3220 then determines {FTs: N} as an unordered combination of the trees enumerated for all allocations A_(i) of N−1 nodes determined at block 3216. In this particular example, {FTs: N} is determined as the union of the sets {FTs: A_(i)} determined at blocks 3208 through 3218.

An example of enumerating a tree from a finite number of nodes and determining a natural numeral for these trees in connection with process 3200 is illustrated with reference to FIGS. 18 through 25 in which the number of nodes for each enumerated tree is N=4. In this particular example, the enumerated trees will be BELTs, defining, for this particular embodiment, the initial conditions to include k=2 and r=1 in the push operation of relation (2). However, this is merely an example of a tree that may have N nodes and the presently illustrated embodiment may be applied to the enumeration of differently labeled trees. For example, the presently illustrated example may be extended to non-BELT trees by selecting different values for “k” and “r” for the push operation of relation (2). However, a push operation other than that of relation (2) may be employed depending on the particular embodiment.

In an initial operation, {FTs: 4}=Ø. In a base operation, {FTs: 1}={1}. A distribution list for {FTs: 4} provides possible allocations of N (here, four) nodes among up to N−1 (here, three) subtree slots is shown in Table 2 below (as determined at block 3206):

TABLE 2 Allocation of N − 1 (Three) Nodes Among up to N − 1 Subtree Slots Subtree Slots Allocation Number 0 1 2 0 3 0 0 1 2 1 0 2 1 1 1

In the allocations of three nodes among subtree slots of Table 2, an allocation may allocate to a subtree slot one, two or all three nodes. Thus, for any of these subtree slots, the process 3200 may enumerate sets of fully configured subtrees for these allocations of one, two or all three nodes to a subtree slot as {FTs: 1} (determined as {1} in the base operation), {FTs: 2} and {FTs: 3}, respectively. Block 3212 may recursively determine values for these sets. For {FTs: 3}, a distribution list setting forth possible allocations of three nodes among two subtree slots is shown in Table 3 below:

TABLE 3 Allocation of Two Nodes Among Two Subtree Slots Subtree Slots Allocation Number 0 1 0 2 0 1 1 1

For {FTs: 2}, there is one unique allocation of a single node depending from a root node in a single subtree slot and provides a subtree as shown in FIG. 18. The push operation is then applied to determine the elements of {FTs: 2} Since the possible subtree structures are BELTs (here, with k=1), {FTs: 2} is expressed as the union of a zero-push and one-push values as follows:

{ < j = 0, k = 1 > −push[{1}]}U{ < j = 1, k = 1 > −push[{1}]} = { < j = 0, k = 1 > −push(1)}U{ < j = 1, k = 1 > −push(1)} = {P(1)UP(2)} = {2, 3}

These enumerated trees in {FTs: 2} are shown in FIG. 19. Since there is only one allocation of nodes to a single subtree slot for {FTs: 2}, there is no merger operation for determining the elements of {FTs: 2}.

To determine the elements of {FTs: 3}, a first component of elements is determined from the allocation of two nodes to a single subtree slot (as shown in allocation 0 of Table 3) and a second component of elements is determined from the allocation of a single node to two subtree slots (as shown in allocation 0 of Table 3). Regarding the first component of {FTs: 3}, these two nodes allocated to a single subtree slot may be formed in four different binary edge labeled strings as illustrated in FIG. 20. These binary edge labeled strings are, in effect, either a zero-push or one-push of the trees enumerated in {FTs: 2}={2, 3}. As such, the set of values for these trees may be enumerated as follows:

{<j=0,k=1>—push[{2,3}]}U{<j=1,k=1>—push[{2,3}]}

{<j=0,k=1>—push(2)}U{<j=1,k=1>—push(2)}U

{<j=0,k=1>—push(3)}U{<j=1,k=1>—push(3)}={5,7,11,13}

The second, remaining component of elements of {FTs: 3}, may be represented in three different BELTs as illustrated in FIG. 21. As such, the set of values for these trees may be enumerated by performing a set merger operation of {FT: 2} (as shown in FIG. 19) with itself as follows (block 3216):

{FT:2}×{FT:2}={2,3}×{2,3}={2*2,2*3,3*3}={4,6,9}

Accordingly, the union of the elements from first and second components of {FT: 3} (as derived above from the allocations 0 and 1 in Table 3) provides {FTs: 3}={4, 5, 6, 7, 9, 11, 13}. As shown in FIG. 22, each of these elements correspond with a unique subtree in {FTs: 3} and is associated with a corresponding natural numeral.

To determine the elements of {FTs: 4}, a first component of the elements is determined from the allocation of three nodes a single subtree slot (as shown in allocation 0 of Table 2), a second component of the elements is determined from the allocation of two nodes to a first subtree slot and one node to a second subtree slot (as shown in allocation 1 of Table 2) and a third component of the elements is determined from the allocation of a single node to three subtree slots (as shown in allocation 2 of Table 2). {FTs: 4} comprises the union of these three components.

The elements of the first component of {FTs: 4}, derived from the allocation of three nodes allocated to a single subtree slot, comprises, in effect, a union of a zero-push and one-push of {FTs: 3} (={4, 5, 6, 7, 9, 11, 13} as derived above and graphically illustrated in FIG. 22). These elements of the first component of {FTs: 4} are depicted graphically in FIG. 25 and are enumerated as follows:

{ < j = 0, k = 1 > −push[{FTs:  3}]}U{ < j = 1, k = 1 > −push[{FTs:  3}]} = { < j = 0, k = 1 > −push[{4, 5, 6, 7, 9, 11, 13}]}U{ < j = 1, k = 1 > −push[{4, 5, 6, 7, 9, 11, 13}]} = { < j = 0, k = 1 > −push(4)}U{ < j = 1, k = 1 > −push(4)}U{ < j = 0, k = 1 > −push(5)}U{ < j = 1, k = 1 > −push(5)}U{ < j = 0, k = 1 > −push(6)}U{ < j = 1, k = 1 > −push(6)}U{ < j = 0, k = 1 > −push(7)}U{ < j = 1, k = 1 > −push(7)}U{ < j = 0, k = 1 > −push(9)}U{ < j = 1, k = 1 > −push(9)}U{ < j = 0, k = 1 > −push(11)}U{ < j = 1, k = 1 > −push(11)}U{ < j = 0, k = 1 > −push(13)}{U < j = 1, k = 1 > −push(13)} = {17, 19, 23, 29, 31, 37, 41, 43, 59, 61, 73, 79, 97, 101}.

The elements of the second component of {FTs: 4} are derived from the allocation of two nodes to a first subtree slot and a single node to a second subtree slot. These elements comprise, in effect, a set merger of {zero-push[{FTs: 2}] U one-push[{FTs: 2}]} (graphically illustrated in FIG. 20) and {zero-push[{FTs: 1}] U one-push[{FTs: 1}]} (graphically illustrated in FIG. 19). The elements from the resulting set merger operation are graphically illustrated in FIG. 23 and the corresponding numbers representing the resulting elements are determined as follows (block 3216):

{ < j = 0, k = 1 > −push[{FTs:  2}]U < j = 1, k = 1 > −push[{FTs:  2}]} × { < j = 0, k = 1 > −push[{FTs:  1}]U < j = 1, k = 1 > −push[{FTs:  1}]} = { < j = 0, k = 1 > −push[{2, 3}]U < j = 1, k = 1 > −push[{2, 3}]} × { < j = 0, k = 1 > −push[{1}]U < j = 1, k = 1 > −push[{1}]} = {5, 7, 11, 13} × {2, 3} = {10, 14, 15, 21, 22, 26, 33, 39}

The elements of the third component of the elements of {FTs: 4} are derived from the allocation of a single node to each of three subtree slots. These elements are, in effect, a set merger operation of {zero-push[{FTs: 1}] U one-push[{FTs: 1}]} (as graphically illustrated in FIG. 19) with itself twice. The elements from the resulting set merger operation are graphically illustrated in FIG. 24 and the corresponding numerals representing the elements are determined as follows (block 3216):

{ < j = 0, k = 1 > −push[{FTs:  1}]U < j = 1, k = 1 > −push[{FTs:  1}]} × { < j = 0, k = 1 > −push[{FTs:  1}]U < j = 1, k = 1 > −push[{FTs:  1}]} × { < j = 0, k = 1 > −push[{FTs:  1}]U < j = 1, k = 1 > −push[{FTs:  1}]} = {2, 3} × [{2, 3} × {2, 3}] = {1, 3} × {4, 6, 9} = {8, 12, 18, 27}

The union of the first, second and third components of the elements of {FTs: 4}, separately derived from allocations 0, 1 and 3 of three nodes among subtree slots (block 3220), is provided as follows:

{FTs:  4} = {17, 19, 23, 29, 31, 37, 41, 43, 59, 61, 73, 79, 97, 101,}U{10, 14, 15, 21, 22, 26, 33, 39}U{8, 12, 18, 27} = {8, 10, 12, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 29, 31, 33, 37, 39, 41, 43, 59, 61, 73, 79, 97, 101}.

While the above illustrated example is a specific case of enumerating BELTs from four nodes, it should be understood that the process 3200 is general enough to enumerate trees for any finite number of nodes N. Also, while the illustrated example is specifically directed to enumerating BELTs, the claimed subject matter is not limited to this specific example.

According to an embodiment, the technique described above may have many applications in the management and/or manipulation of hierarchical data. The ability to enumerate possible fully configured trees from a finite number of nodes may be applied to any one of several database management applications. In one example, with a priori knowledge of a size and type of tree representing hierarchical data in a cache memory, for example, a cache process may anticipate the extent of requests for retrieval of hierarchical data represented by a tree.

In another application, pattern matching may be used as a form of answering queries. Using the above described technique for enumerating possible trees from a finite number of nodes, possible tree patterns may be efficiently generated in advance of receipt of a query to be matched with one of the generated tree patterns. Here, for example, a set of enumerated trees may be generated prior to a query to anticipate all potential hierarchical relationships for a given set of data. The query may then be mapped and matched with all possible tree structures with a given number of nodes. By anticipating all potential hierarchical relationships in advance of the query, the query may be answered more quickly. Of course, these are merely examples and the claimed subject matter is not limited to these examples.

It will, of course, be understood that, although particular embodiments have just been described, the claimed subject matter is not limited in scope to a particular embodiment or implementation. For example, one embodiment may be in hardware, such as implemented to operate on a device or combination of devices, for example, whereas another embodiment may be in software. Likewise, an embodiment may be implemented in firmware, or as any combination of hardware, software, and/or firmware, for example. Likewise, although the claimed subject matter is not limited in scope in this respect, one embodiment may comprise one or more articles, such as a storage medium or storage media. This storage media, such as, one or more CD-ROMs and/or disks, for example, may have stored thereon instructions, that when executed by a system, such as a computer system, computing platform, or other system, for example, may result in an embodiment of a method in accordance with the claimed subject matter being executed, such as one of the embodiments previously described, for example. As one potential example, a computing platform may include one or more processing units or processors, one or more input/output devices, such as a display, a keyboard and/or a mouse, and/or one or more memories, such as static random access memory, dynamic random access memory, flash memory, and/or a hard drive, although, again, the claimed subject matter is not limited in scope to this example.

In the preceding description, various aspects of the claimed subject matter have been described. For purposes of explanation, specific numbers, systems and/or configurations were set forth to provide a thorough understanding of the claimed subject matter. However, it should be apparent to one skilled in the art having the benefit of this disclosure that the claimed subject matter may be practiced without the specific details. In other instances, well-known features were omitted and/or simplified so as not to obscure the claimed subject matter. While certain features have been illustrated and/or described herein, many modifications, substitutions, changes and/or equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and/or changes as fall within the true spirit of the claimed subject matter. 

1. A method comprising: enumerating possible trees configurable from a finite number (N) of nodes; and determining for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals.
 2. The method of claim 1, wherein the method further comprises: determining one or more allocations of the finite number of nodes among subtree slots, an allocation defining a portion of the finite number of nodes to a subtree slot in the allocation; for a subtree slot in an allocation, enumerating a set containing one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.
 3. The method of claim 2, wherein the method further comprises pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot.
 4. The method of claim 3, wherein the method further comprises merging pushed sets of subtrees in an allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.
 5. The method of claim 4, wherein the method further comprises determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from the allocations of the finite number of nodes among subtree slots.
 6. The method of claim 4, the method further comprising: representing a subtree in a pushed set of subtrees of the allocation with a natural numeral; and multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with a natural numeral corresponding to an element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.
 7. The method of claim 4, the method further comprising multiplying a natural numeral corresponding to an element in the pushed set of subtrees in the allocation with an element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.
 8. The method of claim 1, wherein the enumerated trees comprise binary edge labeled trees.
 9. The method of claim 1, the method further comprising: identifying N−1 arrangements of subtree slots coupled to a root node; and determining possible allocations of N−1 nodes among subtree slots in arrangements of subtree slots; for a subtree slot in a possible allocation of the N−1 nodes, determining one or more natural numerals for possible configurations of a subtree from nodes allocated to the subtree slot.
 10. The method of claim 9, the method further comprising: allocating a portion of N−1 nodes to a subtree slot in an arrangement of subtree slots; and for a subtree slot in the arrangement of subtree slots: enumerating one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot; and performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot.
 11. The method of claim 1, wherein said enumerating comprises enumeration of each of one or more possible trees, and wherein said determining a natural numeral for the enumerated trees comprises determining a natural numeral for each one of the enumerated trees.
 12. The method of claim 11, the method further comprising: identifying each of N−1 arrangements of subtree slots coupled to a root node; determining each possible allocation of N−1 nodes among subtree slots in each arrangement of subtree slots; for each subtree slot in each possible allocation of the N−1 nodes, determining one or more natural numerals for each possible configuration of a subtree from nodes allocated to the subtree slot.
 13. The method of claim 12, the method further comprising: allocating a portion of N−1 nodes to each subtree slot in each arrangement of subtree slots; and for each subtree slot in the arrangement of subtree slots: enumerating each of one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot; and performing a push operation on the enumerated one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot to determine one or more natural numerals, each natural numeral being associated with a corresponding one of said pushed one or more possible subtrees configurable from the portion of the N−1 nodes allocated to the subtree slot.
 14. The method of claim 11, wherein the method further comprises: determining one or more allocations of the finite number of nodes among subtree slots, each allocation defining a portion of the finite number of nodes to a subtree slot in the allocation; for each subtree slot in each allocation, enumerating a set containing each of one or more possible subtrees configurable from the portion of the finite number of nodes allocated to the subtree slot.
 15. The method of claim 14, wherein the method further comprises: pushing the elements of the enumerated set of subtrees to generate a pushed set corresponding with the subtree slot; and merging pushed sets of subtrees in each allocation of the finite number of nodes to subtree slots to determine a set of trees that are configurable from the allocation.
 16. The method of claim 15, wherein the method further comprises multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each element of all other sets of pushed subtrees in the allocation to determine the one or more natural numerals corresponding with the at least some of the enumerated trees.
 17. The method of claim 15, the method further comprising: representing each subtree in each pushed set of subtrees of the allocation with a natural numeral; and multiplying each natural numeral corresponding to an element in the pushed set of subtrees in the allocation with each natural numeral corresponding to each element of one or more other sets of pushed subtrees in the allocation to determine one or more natural numerals corresponding with at least some of the enumerated trees.
 18. The method of claim 15, wherein the method further comprises determining the one or more possible trees configurable from the finite number of nodes as the union of trees which are configurable from each of the allocations of the finite number of nodes among subtree slots. 19.-56. (canceled)
 57. An apparatus comprising a computing platform, the computing platform being adapted to: enumerate possible trees configurable from a finite number (N) of nodes; and determine for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals. 58.-84. (canceled)
 85. An article comprising: a storage medium comprising machine-readable instructions stored thereon to: enumerate possible trees configurable from a finite number (N) of nodes; and determine for the enumerated trees natural numerals associated with particular ones of the enumerated trees, the natural numerals being based, at least in part, on a predetermined association between trees and natural numerals. 86.-112. (canceled) 